Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| isoclinism | isoclinic class of Dic26 |
| automorphism | Aut(G) = C23×AGL(1,𝔽13) (order 1248) · Inn(G) = C2×D13 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×C22⋊D13 |
| Fitting | C22×C3×C13 ⋊ C2 |
| Hall | C3×C13 ⋊ D4 |
| Sylow | C13 ⋊ C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D13 |
| commutator, cosocle | C2×C13 | C22×C3 |
| Frattini | C2 | C2×C3×D13 |
| Fitting | C22×C3×C13 | C2 |
| socle | C2×C3×C13 | C22 |
| hypercenter | C22×C3 | D13 |
| hypocenter | C13 | C3×D4 |
| LCS | G ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 156 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,4,6,8,10,12,13,11,9,7,5,3)(15,17)(18,19,20);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)87 conjugacy classes — view the full character table.
| maximal subgroup of | #10597745, #10597746, D3×C22⋊D13, … |
| maximal normal subgroup of | #10597745, #10597746, D3×C22⋊D13, … |
| maximal quotient of | C3×#6101, C3×#6103, C3×#6104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 31);